{"id":"n-89a0af04eae7a94e","revision":0,"first_index":0,"fingerprint":"f6-49176c5aecdbb25b","name":"theta_plus_one of veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[[],["-1"],["2"],["-3","-2"],["4","8"],["-5","-20","-3"],["6","40","18"],["-7","-70","-63","-4"],["8","112","168","32"],["-9","-168","-378","-144","-5"],["10","240","756","480","50"],["-11","-330","-1386","-1320","-275","-6"],["12","440","2376","3168","1100","72"],["-13","-572","-3861","-6864","-3575","-468","-7"],["14","728","6006","13728","10010","2184","98"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","-1","2","-5","12","-28","64","-144","320","-704","1536","-3328","7168","-15360","32768"],"nonn":false,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["-1"]],"recurrence":{"denominator":{"coeffs":[["2","0"],["-2","1"],["0","-2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-8","0"],["2","4","2"],["0","1","1"],["0","-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4","0"],["0","0","-2","2"],["0","0","-2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171435,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-d2c52d059b6d5ac4"]},{"id":171436,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-d2c52d059b6d5ac4"]},{"id":171437,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-d2c52d059b6d5ac4"]}],"region_exact":{},"incoming_edges":[{"id":"e-d2c52d059b6d5ac4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-89a0af04eae7a94e","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"5cefa2556f4d0701b5253ba0d35b7d9b9ee1da9f9157861ceb1a2268c63ab295","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a75381dbce977094","role":"sequence","name":"veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1e5b73b8d1a3020f","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-8e043fdc31547ace","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-8e043fdc31547ace","other":{"id":"n-8e043fdc31547ace","name":"theta_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["20"]],"card_rows":[["4"],["20"],["60","40"],["140","280"]],"row_sums_preview":["0","4","20","100","420","1568","5376","17280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-4dba393820f57c76","e-043949eb589e063a"],"text":"A178820 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A178820 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-4237ceb69505b4a1","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-54a374793669f817","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-54a374793669f817","other":{"id":"n-54a374793669f817","name":"theta_plus_one of veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["12","8"],["20","40"]],"row_sums_preview":["0","0","2","6","20","60","168","448"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-888e3b1c2eb8e007","e-d2c1d39a7dcf5179"],"text":"A003506 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A003506 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-6094b022c4c5048f","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-11af5ed75ed92b6e","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-11af5ed75ed92b6e","other":{"id":"n-11af5ed75ed92b6e","name":"theta_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["42"]],"card_rows":[["6"],["42"],["168","112"],["504","1008"]],"row_sums_preview":["0","6","42","280","1512","7056","29568","114048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-d44b85edec7c5853","e-5a0fe12d2e98d462"],"text":"A178822 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A178822 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-97c3bc0464b5ac2a","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-dd944ac9aa313a2a","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-dd944ac9aa313a2a","other":{"id":"n-dd944ac9aa313a2a","name":"theta_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["30"]],"card_rows":[["5"],["30"],["105","70"],["280","560"]],"row_sums_preview":["0","5","30","175","840","3528","13440","47520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-1471988cf5f0d7aa","e-0a6c66d84e647469"],"text":"A178821 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A178821 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-aead77a770a38a89","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-a5cadb38ec0f6a2b","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-a5cadb38ec0f6a2b","other":{"id":"n-a5cadb38ec0f6a2b","name":"theta_plus_one of veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8"]],"card_rows":[["2"],["8"],["24","16"],["64","128"]],"row_sums_preview":["0","2","8","40","192","896","4096","18432"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-0d347b78871c0570","e-825ce42461bb84ad"],"text":"A038208 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A038208 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-afecfca317864f01","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-2044aa2914d76bfa","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-2044aa2914d76bfa","other":{"id":"n-2044aa2914d76bfa","name":"theta_plus_one of veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","4"],["8","16"]],"row_sums_preview":["0","2","4","10","24","56","128","288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-53a911614b54e844","e-1dae91a8e46308bd"],"text":"A028326 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A028326 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-b0892b03b28af1b3","kind":"fingerprint_related","node_a_id":"n-032bb528bb3136f5","node_b_id":"n-89a0af04eae7a94e","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-032bb528bb3136f5","other":{"id":"n-032bb528bb3136f5","name":"veronese_section_2_1 of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","4"],["8","16"]],"row_sums_preview":["0","2","4","10","24","56","128","288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-801895951aa20f29","steps":["Veronese 2 (j=1)"],"edges":["e-274dee66a9035b75"],"text":"A103406 \u2192 Veronese 2 (j=1)"},"headline":"A103406 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-ca776e25b09aeb4b","kind":"fingerprint_related","node_a_id":"n-89a0af04eae7a94e","node_b_id":"n-84017662cec584dd","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-84017662cec584dd","other":{"id":"n-84017662cec584dd","name":"theta_plus_one of Triangle of odd-numbered terms in rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["3","2"]],"card_rows":[["1"],["2"],["3","2"],["4","8"]],"row_sums_preview":["1","2","5","12","28","64","144","320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fd133d9d2402ed18","steps":["\u03b8+1"],"edges":["e-baa92bf22b9daffc"],"text":"A034867 \u2192 \u03b8+1"},"headline":"A034867 \u2192 \u03b8+1","chain":null}},{"id":"l-f3cd02cb7ed3c0f8","kind":"fingerprint_related","node_a_id":"n-478a0e862ab5a96b","node_b_id":"n-89a0af04eae7a94e","fingerprint":"f6-49176c5aecdbb25b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-49176c5aecdbb25b"},"other_id":"n-478a0e862ab5a96b","other":{"id":"n-478a0e862ab5a96b","name":"derivative of veronese_section_2_1 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-49176c5aecdbb25b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["3","2"],["4","8"]],"row_sums_preview":["0","1","2","5","12","28","64","144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["Veronese 2 (j=1)","D"],"edges":["e-6fdf1329e5e61ed8","e-7d22e275be455815"],"text":"A159854 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A159854 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  0;\n  1: -1;\n  2:  2;\n  3: -3,   -2;\n  4:  4,    8;\n  5: -5,  -20,   -3;\n  6:  6,   40,   18;\n  7: -7,  -70,  -63,   -4;\n  8:  8,  112,  168,   32;\n  9: -9, -168, -378, -144, -5;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-2-(n^3-2*n^2+n)*x)*P(n,x) = (2*n-(n^3-n^2-4*n+8)*x-(n^3-n^2-2*n)*x^2)*P(n-1,x) + (4*x-(2*n^2+2*n+4)*x^2+(2*n^2+2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - (n^3 - n^2 - 4*n + 8)*x - (n^3 - n^2 - 2*n)*x^2)*P[n - 1, x] + (4*x - (2*n^2 + 2*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*D[P[n - 1, x], x])/(-2*n + 2 + (n^3 - 2*n^2 + n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - (n^3 - n^2 - 4*n + 8)*x - (n^3 - n^2 - 2*n)*x^2)*v[i - 1] + (4*x - (2*n^2 + 2*n + 4)*x^2 + (2*n^2 + 2*n)*x^3)*deriv(v[i - 1], x))/(-2*n + 2 + (n^3 - 2*n^2 + n)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-a70814c31408551f","e-d2c52d059b6d5ac4"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}